Lecture 3: Adaptive Construction of Response Surface Approximations for Bayesian Inference
|
|
- Bertram Douglas
- 6 years ago
- Views:
Transcription
1 Lecture 3: Adaptive Construction of Response Surface Approximations for Bayesian Inference Serge Prudhomme Département de mathématiques et de génie industriel Ecole Polytechnique de Montréal SRI Center for Uncertainty Quantification King Abdullah University of Science and Technology evita Winter School 2015 Uncertainty Quantification for Physical Phenomena Geilo, Norway, January 18-23, 2015 S. Prudhomme Adaptive Response Approximations January 18-23, / 28
2 Outline Outline Introduction. A few words about validation. Application of GOEE to Bayesian Inference. Numerical examples. S. Prudhomme Adaptive Response Approximations January 18-23, / 28
3 Introduction Introduction Questions pertaining to validation Is the model any good? Is the model a good model? How good is the model? If one wants to be philosophical... What is a good model? Before answering those questions, one must have an objective in mind. Quantities of interest: Specific objectives that can be expressed as the target outputs of a model (mathematically, they are often defined by functionals of the solutions). Examples: Q(u) = u(x) Q(u) = u(x)dx S. Prudhomme Adaptive Response Approximations January 18-23, / 28
4 Introduction Some Definitions: Verification: The process of determining the accuracy with which a computational model can produce results deliverable by the mathematical model on which it is based. Code and Solution Verification Validation: The process of determining the accuracy with which a model can predict observed physical events (or the important features of a physical reality). P. Roache (2009): The process of determining the degree to which a model (and its associated data) is an accurate representation of the real world from the perspective of the intended uses of the model. Uncertainty Quantification: The process of determining the degree of uncertainty in the prediction of the QoI. Typically, the degree of uncertainty is related to the probability distribution for the QoI. S. Prudhomme Adaptive Response Approximations January 18-23, / 28
5 Introduction Paths to Knowledge DECISION THE UNIVERSE of PHYSICAL REALITIES KNOWLEDGE Observational Errors Modeling Errors Discretization Errors OBSERVATIONS VALIDATION THEORY / MATHEMATICAL MODELS VERIFICATION COMPUTATIONAL MODELS Oden, Moser, and Ghattas, SIAM News, (Nov. 2010) Oden and Prudhomme, IJNME (Sept. 2010) S. Prudhomme Adaptive Response Approximations January 18-23, / 28
6 Introduction Control of Errors Errors are all a matter of comparison! Code Verification: Using the method of manufactured solutions, for example, we can easily compare the computed solution with the manufactured solution. Solution Verification: In this case, the solution of the problem is unknown and one can use convergence (uniform or adaptive methods) to assess the accuracy of the approximate solutions. Calibration Process: Comparison of observable data with model estimates of the observables. Validation Process: The main idea behind validation is to know whether a model can be used for prediction purposes. What should we compare in this case? S. Prudhomme Adaptive Response Approximations January 18-23, / 28
7 Validation Process Validation Process 1. Calibration: Identification of values of parameters of a model designed to bring model results into agreement with measurements. 2. Validation: The process of determining the accuracy with which a model can predict observed physical events (or the important features of a physical reality). 3. Prediction: The forecast of an event (a predicted event cannot be measured or observed, for then it ceases to be a prediction). S. Prudhomme Adaptive Response Approximations January 18-23, / 28
8 Validation Process The Prediction Pyramid for validating a single model If model predicts QoI within prescribed accuracy the model is said to be not invalidated Prediction of QoI Q P S P Prediction Scenario No Data Available S V Validation Scenario(s) (more complex than calibration) S C Calibration Scenarios (As simple as possible, numerous) Experimental Data D V on validation scenario(s) Experimental Data D C on calibration scenarios The Validation Pyramid S. Prudhomme Adaptive Response Approximations January 18-23, / 28
9 Validation Process Classical Approach for Validation Calibration Data D c? Model with Unknown Parameter(s) θ Solve Inverse Problem on Calibration Scenario Model with Calibrated Parameter(s) θ c Solve Forward Problem on Validation Scenario Estimate Observables O c Model is Invalid No M ( Oc, D v ) < γ Validation Data D v D v Yes Model not Invalid Increased Confidence M ( O cal, D val ) γtol (?) S. Prudhomme Adaptive Response Approximations January 18-23, / 28
10 Validation Process Proposed Validation Process (2009) Model with Unknown Parameter(s) θ Same Model with Parameter(s) θ Calibration Data Dc Solve Inverse Problem on Calibration Scenario Solve Inverse Problem on Validation Scenario Validation Data Dv Model with Calibrated Parameter(s) θ c θ v Model with Re Calibrated Parameter(s) Estimate QoI Solving the Forward Problem on Prediction Scenario Sensitivity/ Uncertainty Quantification M ( Q cal, Q val ) γtol (?) Qc Q v? Model is Invalid No M ( Qc, Q v ) < γ Yes Model not Invalid Increased Confidence S. Prudhomme Adaptive Response Approximations January 18-23, / 28
11 Planning of Validation Processes Validation process requires detailed planning: 1. Description of goals: Describe background and goals of the predictions. Clearly define the quantity (or quantities) of interest. 2. Modeling: Write mathematical equations of selected model(s), list all parameters that are necessary to solve the problem, as well as assumptions and limitations of the model(s), 3. Data collection: Collect as many data as possible from literature or available sources (data should include, if available, the statistics). 4. Sensitivity analysis: Quantify the sensitivity of QoI with respect to parameters of the model. Rank parameters according to their influence. 5. Calibration experiments: Provide description of scenario (as precisely as possible), observables and statistics, prior and likelihood of the parameters to be calibrated. 6. Validation experiments: Provide same as above + clearly state assumption to be validated. S. Prudhomme Adaptive Response Approximations January 18-23, / 28
12 Planning of Validation Processes Morgan & Henrion s Ten Commandements (1990) In relation to quantitative risk and policy analysis 1. Do your homework with literature, experts and users. 2. Let the problem drive the analysis. 3. Make the analysis as simple as possible, but no simpler. 4. Identify all significant assumptions. 5. Be explicit about decision criteria and policy strategies. 6. Be explicit about uncertainties. 7. Perform systematic sensitivity and uncertainty analysis. 8. Iteratively refine the problem statement and the analysis. 9. Document clearly and completely. 10. Expose to peer review. Extracted from D. Vose, Risk Analysis: A Quantitative Guide (2008) S. Prudhomme Adaptive Response Approximations January 18-23, / 28
13 Planning of Validation Processes A systematic approach to the planning and implementation of experiments (Chapter 1 - Section 2) In Wu & Hamada Experiments, Analysis, and Optimization (2009) 1. State objective. 2. Choose response. 3. Choose factors and levels. 4. Choose experimental plan. 5. Perform the experiment. 6. Analyze the data. 7. Draw conclusions and make recommendations:... the conclusions should refer back to the stated objectives of the experiment. A confirmation experiment is worthwhile for example, to confirm the recommended settings. Recommendations for further experimentation in a follow-up experiment may also be given. For example, a follow-up experiment is needed if two models explain the experimental data equally well and one must be chosen for optimization. S. Prudhomme Adaptive Response Approximations January 18-23, / 28
14 Planning of Validation Processes Planning Planning is a cumbersome and time-consuming process. Planning of validation processes involves many choices that eventually need to be carefully checked. Choices are made about: Physical models Quantities of interest and surrogate quantities of interest Experiments for calibration and validation purposes Data sets to be used in calibration and validation Prior pdf and likelihood function Probabilistic models... Our preliminary experiences with validation has revealed that many sanity checks need to be added within the proposed validation process. Our objective is to develop a suite of tools to systematically verify the correctness of each stage of the validation process. S. Prudhomme Adaptive Response Approximations January 18-23, / 28
15 Efficient Bayesian inference - model selection for RANS Adaptive Response Surface for Parameter Estimation Objective The main objective here is to develop a methodology based on response surface models and goal-oriented error estimation for efficient and reliable parameter estimation in turbulence modeling. S. Prudhomme Adaptive Response Approximations January 18-23, / 28
16 Efficient Bayesian inference - model selection for RANS Bayesian inference for RANS using surrogate models Efficient Bayesian inference: Approximate response surface models can be used to reduce the computational cost of the process. Ma and Zabaras, Li and Marzouk, 2014; Marzouk and Xiu, 2009; Marzouk and Najm, UQ for RANS models: Uncertainty in the RANS model parameters is a known issue in the turbulence community, but quantifying the effect of this uncertainty is seldom analyzed in the computational fluid dynamics literature. Cheung et al., Oliver and Moser, S. Prudhomme Adaptive Response Approximations January 18-23, / 28
17 Efficient Bayesian inference - model selection for RANS Fully developed incompressible channel flow Mean flow equations: u = U + u DU i Dt = 1 P + ρ x i x j U = 0 Eddy viscosity assumption: u i u j = ν T (U i,j + U j,i ) ( ν U ) i u i x u j j Channel equations: assuming homogeneous turbulence in x ( (ν + ν T ) U ) = 1, y (0, H) y y 1 Durbin and Petterson Reif, 2001; Pope, 2000 S. Prudhomme Adaptive Response Approximations January 18-23, / 28
18 Efficient Bayesian inference - model selection for RANS Spalart-Allmaras (SA) model Eddy viscosity is given by: ν T = νf v1, f v1 = χ 3 /(χ 3 + c v1 3 ), χ = ν/ν where ν is governed by the transport equation with D ν Dt = P ν(κ, c b1 ) ε ν (κ, c b1, σ SA, c w2 ) + 1 [ σ SA x j P ν = production term D ν = wall destruction term ( (ν + ν) ν x j ) ] ν ν + c b2 x j x j Parameter Values κ 0.41 c b c b c v1 7.1 σ SA 2/3 c w Allmaras, Johnson, and Spalart, 2012; Oliver and Darmofal, 2009 S. Prudhomme Adaptive Response Approximations January 18-23, / 28
19 Efficient Bayesian inference - model selection for RANS Forward model: Find U and ν such that 1 = ( (ν + ν T (c v1 )) U ) y y 0 = P ν (κ, c b1 ) ε ν (κ, c b1, σ SA, c w2 ) + 1 σ SA Boundary conditions: Weak formulation: y [ ( (ν + ν) ν ) ( ) ] 2 ν + c b2 y y U(0) = 0, y U(H) = 0, ν(0) = 0, y ν(h) = 0 Find (U, ν) V s.t. B((U, ν); (V, µ)) = F(V, µ), (V, µ) V Quantity of interest and adjoint problem: Find (Z, ζ) V s.t. B ((U, ν); (Z, ζ), (V, µ)) = Q(V, µ) = H 0 V dy (V, µ) V S. Prudhomme Adaptive Response Approximations January 18-23, / 28
20 Efficient Bayesian inference - model selection for RANS Adaptive Response Surface Uniform priors for all parameters with range (0.5, 1.5) times nominal value (e.g. κ U(0.205, 0.615)) Adapted expansion order (after 17 iterations): κ c b1 σ SA c b2 c v1 c w E L 2 (Ξ) adaptive uniform 0.14 surrogate full model Number of evaluations Q S. Prudhomme Adaptive Response Approximations January 18-23, / 28
21 Efficient Bayesian inference - model selection for RANS Bayesian inference Bayes rule: p(ξ q) = L(ξ q) p(ξ) p(q) where q R n L(ξ q) p(ξ) p(ξ q) = Calibration data = Likelihood = Prior = Posterior Model selection: Set of models M = {M 1, M 2,..., M n } Posterior plausibility = p(m i q, M) Likelihood = E(M i q, M) := p(q M i, M) = Ξ p(q ξ, M i, M)p(ξ M i, M) dξ p(m i q, M) E(M i q, M) p(m i M) 1 Calvetti and Somersalo, 2007; Jaynes, 2003; Kaipio and Somersalo, 2005 S. Prudhomme Adaptive Response Approximations January 18-23, / 28
22 Efficient Bayesian inference - model selection for RANS Calibration data: Data is obtained from direct numerical simulation (DNS) 1 Mean velocity measurements were taken at Re τ = 944 and Re τ = 2003 Uncertainty models: Three multiplicative error models u + (z; ξ) = (1 + ɛ(z; ξ))u + (z; ξ) independent homogeneous covariance correlated homogeneous covariance correlated inhomogeneous covariance Reynolds stress model u i u + j (z; ξ) = T + (z; ξ) ɛ(z; ξ) 1 Del Alamo et al., 2004; Hoyas and Jiménez, 2006 S. Prudhomme Adaptive Response Approximations January 18-23, / 28
23 Efficient Bayesian inference - model selection for RANS Numerical Results Independent homogeneous covariance: u + (z; ξ) = (1 + ɛ(z; ξ))u + (z; ξ) ɛ(z)ɛ(z ) = σ 2 δ(z z ) κ c v σ S. Prudhomme Adaptive Response Approximations January 18-23, / 28
24 Efficient Bayesian inference - model selection for RANS Numerical Results Correlated homogeneous covariance: u + (z; ξ) = (1 + ɛ(z; ξ))u + (z; ξ) ɛ(z)ɛ(z ) = σ 2 exp ( 1/2 (z z ) 2 ) l κ c v σ S. Prudhomme Adaptive Response Approximations January 18-23, / 28
25 Efficient Bayesian inference - model selection for RANS Numerical Results Correlated inhomogeneous covariance: u + (z; ξ) = (1 + ɛ(z; ξ))u + (z; ξ) ɛ(z)ɛ(z ) ( 2l(z)l(z = σ 2 ) ) 1/2 l 2 (z) + l 2 (z exp ( (z z ) 2 ) ) l 2 (z) + l 2 (z ) κ c v1 σ S. Prudhomme Adaptive Response Approximations January 18-23, / 28
26 Efficient Bayesian inference - model selection for RANS Numerical Results Reynolds stress uncertainty: u i u + j (z; ξ) = T + (z; ξ) ɛ(z; ξ) ɛ(z)ɛ(z ) = k in (z, z ) + k out (z, z ) where k in models the error near the walls and k out far from the walls κ c v σ in S. Prudhomme Adaptive Response Approximations January 18-23, / 28
27 Efficient Bayesian inference - model selection for RANS Numerical Results: Model selection Model evidence (log(e)): Surrogate Full model Independent homogeneous Correlated homogeneous Correlated inhomogeneous Reynolds stress Relative runtimes (in seconds): Surrogate Full model Independent homogeneous Correlated homogeneous Correlated inhomogeneous Reynolds stress Cumulative S. Prudhomme Adaptive Response Approximations January 18-23, / 28
28 Conclusions Concluding remarks and future work We need to think about errors and error control in computational science and engineering. We need quantitative methods to assess the reliability of our predictions (with respect to a given goal). Formulation of the problem should be done with respect to the goal of the computation, not necessarily by minimizing the energy of the system. S. Prudhomme Adaptive Response Approximations January 18-23, / 28
Verification and Validation in Computational Engineering and Sciences
Verification and Validation in Computational Engineering and Sciences Serge Prudhomme Département de mathématiques et de génie industriel Ecole Polytechnique de Montréal, Canada SRI Center for Uncertainty
More informationAdaptive surrogate modeling for response surface approximations with application to bayesian inference
Prudhomme and Bryant Adv. Model. and Simul. in Eng. Sci. 2015 2:22 DOI 10.1186/s40323-015-0045-5 RESEARCH ARTICLE Open Access Adaptive surrogate modeling for response surface approximations with application
More informationPredictive Engineering and Computational Sciences. Research Challenges in VUQ. Robert Moser. The University of Texas at Austin.
PECOS Predictive Engineering and Computational Sciences Research Challenges in VUQ Robert Moser The University of Texas at Austin March 2012 Robert Moser 1 / 9 Justifying Extrapolitive Predictions Need
More informationComputational Fluid Dynamics 2
Seite 1 Introduction Computational Fluid Dynamics 11.07.2016 Computational Fluid Dynamics 2 Turbulence effects and Particle transport Martin Pietsch Computational Biomechanics Summer Term 2016 Seite 2
More informationEngineering. Spring Department of Fluid Mechanics, Budapest University of Technology and Economics. Large-Eddy Simulation in Mechanical
Outline Geurts Book Department of Fluid Mechanics, Budapest University of Technology and Economics Spring 2013 Outline Outline Geurts Book 1 Geurts Book Origin This lecture is strongly based on the book:
More informationUncertainty Quantification and Calibration of the k ǫ turbulence model
Master of Science Thesis Uncertainty Quantification and Calibration of the k ǫ turbulence model H.M. Yıldızturan 5-08-202 Faculty of Aerospace Engineering Delft University of Technology Uncertainty Quantification
More informationConcepts and Practice of Verification, Validation, and Uncertainty Quantification
Concepts and Practice of Verification, Validation, and Uncertainty Quantification William L. Oberkampf Sandia National Laboratories (retired) Consulting Engineer wloconsulting@gmail.com Austin, Texas Exploratory
More informationVerification and Validation in Simulations of Complex Engineered Systems
Verification and Validation in Simulations of Complex Engineered Systems Robert D. Moser Todd Oliver, Onkar Sahni, Gabriel Terejanu Institute for Computational Engineering and Sciences The University of
More informationBayesian parameter estimation in predictive engineering
Bayesian parameter estimation in predictive engineering Damon McDougall Institute for Computational Engineering and Sciences, UT Austin 14th August 2014 1/27 Motivation Understand physical phenomena Observations
More informationCHAPTER 11: REYNOLDS-STRESS AND RELATED MODELS. Turbulent Flows. Stephen B. Pope Cambridge University Press, 2000 c Stephen B. Pope y + < 1.
1/3 η 1C 2C, axi 1/6 2C y + < 1 axi, ξ > 0 y + 7 axi, ξ < 0 log-law region iso ξ -1/6 0 1/6 1/3 Figure 11.1: The Lumley triangle on the plane of the invariants ξ and η of the Reynolds-stress anisotropy
More informationAlgorithms for Uncertainty Quantification
Algorithms for Uncertainty Quantification Lecture 9: Sensitivity Analysis ST 2018 Tobias Neckel Scientific Computing in Computer Science TUM Repetition of Previous Lecture Sparse grids in Uncertainty Quantification
More informationPredictive Engineering and Computational Sciences. Full System Simulations. Algorithms and Uncertainty Quantification. Roy H.
PECOS Predictive Engineering and Computational Sciences Full System Simulations Algorithms and Uncertainty Quantification Roy H. Stogner The University of Texas at Austin October 12, 2011 Roy H. Stogner
More informationSequential Importance Sampling for Rare Event Estimation with Computer Experiments
Sequential Importance Sampling for Rare Event Estimation with Computer Experiments Brian Williams and Rick Picard LA-UR-12-22467 Statistical Sciences Group, Los Alamos National Laboratory Abstract Importance
More informationEngineering. Spring Department of Fluid Mechanics, Budapest University of Technology and Economics. Large-Eddy Simulation in Mechanical
Outline Department of Fluid Mechanics, Budapest University of Technology and Economics Spring 2011 Outline Outline Part I First Lecture Connection between time and ensemble average Ergodicity1 Ergodicity
More informationThe Effect of the DNS Data Averaging Time on the Accuracy of RANS-DNS Simulations
The Effect of the DNS Data Averaging Time on the Accuracy of RANS-DNS Simulations Svetlana V. Poroseva 1, The University of New Mexico, Albuquerque, New Mexico, 87131 Elbert Jeyapaul 2, Scott M. Murman
More informationOpenFOAM selected solver
OpenFOAM selected solver Roberto Pieri - SCS Italy 16-18 June 2014 Introduction to Navier-Stokes equations and RANS Turbulence modelling Numeric discretization Navier-Stokes equations Convective term {}}{
More informationBayesian Reliability Analysis: Statistical Challenges from Science-Based Stockpile Stewardship
: Statistical Challenges from Science-Based Stockpile Stewardship Alyson G. Wilson, Ph.D. agw@lanl.gov Statistical Sciences Group Los Alamos National Laboratory May 22, 28 Acknowledgments Christine Anderson-Cook
More informationBayesian Inference: What, and Why?
Winter School on Big Data Challenges to Modern Statistics Geilo Jan, 2014 (A Light Appetizer before Dinner) Bayesian Inference: What, and Why? Elja Arjas UH, THL, UiO Understanding the concepts of randomness
More informationStochastic Collocation Methods for Polynomial Chaos: Analysis and Applications
Stochastic Collocation Methods for Polynomial Chaos: Analysis and Applications Dongbin Xiu Department of Mathematics, Purdue University Support: AFOSR FA955-8-1-353 (Computational Math) SF CAREER DMS-64535
More informationTurbulent drag reduction by streamwise traveling waves
51st IEEE Conference on Decision and Control December 10-13, 2012. Maui, Hawaii, USA Turbulent drag reduction by streamwise traveling waves Armin Zare, Binh K. Lieu, and Mihailo R. Jovanović Abstract For
More informationBayesian Calibration of Simulators with Structured Discretization Uncertainty
Bayesian Calibration of Simulators with Structured Discretization Uncertainty Oksana A. Chkrebtii Department of Statistics, The Ohio State University Joint work with Matthew T. Pratola (Statistics, The
More informationOn a Data Assimilation Method coupling Kalman Filtering, MCRE Concept and PGD Model Reduction for Real-Time Updating of Structural Mechanics Model
On a Data Assimilation Method coupling, MCRE Concept and PGD Model Reduction for Real-Time Updating of Structural Mechanics Model 2016 SIAM Conference on Uncertainty Quantification Basile Marchand 1, Ludovic
More informationScalable Algorithms for Optimal Control of Systems Governed by PDEs Under Uncertainty
Scalable Algorithms for Optimal Control of Systems Governed by PDEs Under Uncertainty Alen Alexanderian 1, Omar Ghattas 2, Noémi Petra 3, Georg Stadler 4 1 Department of Mathematics North Carolina State
More informationICES REPORT A posteriori error control for partial differential equations with random data
ICES REPORT 13-8 April 213 A posteriori error control for partial differential equations with random data by Corey M. Bryant, Serge Prudhomme, and Timothy Wildey The Institute for Computational Engineering
More informationModelling of turbulent flows: RANS and LES
Modelling of turbulent flows: RANS and LES Turbulenzmodelle in der Strömungsmechanik: RANS und LES Markus Uhlmann Institut für Hydromechanik Karlsruher Institut für Technologie www.ifh.kit.edu SS 2012
More informationUncertainty quantification for inverse problems with a weak wave-equation constraint
Uncertainty quantification for inverse problems with a weak wave-equation constraint Zhilong Fang*, Curt Da Silva*, Rachel Kuske** and Felix J. Herrmann* *Seismic Laboratory for Imaging and Modeling (SLIM),
More informationarxiv: v2 [math.na] 6 Aug 2018
Bayesian model calibration with interpolating polynomials based on adaptively weighted Leja nodes L.M.M. van den Bos,2, B. Sanderse, W.A.A.M. Bierbooms 2, and G.J.W. van Bussel 2 arxiv:82.235v2 [math.na]
More informationUncertainty Quantification for Inverse Problems. November 7, 2011
Uncertainty Quantification for Inverse Problems November 7, 2011 Outline UQ and inverse problems Review: least-squares Review: Gaussian Bayesian linear model Parametric reductions for IP Bias, variance
More informationTurbulent eddies in the RANS/LES transition region
Turbulent eddies in the RANS/LES transition region Ugo Piomelli Senthil Radhakrishnan Giuseppe De Prisco University of Maryland College Park, MD, USA Research sponsored by the ONR and AFOSR Outline Motivation
More informationTurbulence Modeling I!
Outline! Turbulence Modeling I! Grétar Tryggvason! Spring 2010! Why turbulence modeling! Reynolds Averaged Numerical Simulations! Zero and One equation models! Two equations models! Model predictions!
More informationComputation for the Backward Facing Step Test Case with an Open Source Code
Computation for the Backward Facing Step Test Case with an Open Source Code G.B. Deng Equipe de Modélisation Numérique Laboratoire de Mécanique des Fluides Ecole Centrale de Nantes 1 Rue de la Noë, 44321
More informationA Computational Investigation of a Turbulent Flow Over a Backward Facing Step with OpenFOAM
206 9th International Conference on Developments in esystems Engineering A Computational Investigation of a Turbulent Flow Over a Backward Facing Step with OpenFOAM Hayder Al-Jelawy, Stefan Kaczmarczyk
More informationINVERSE PROBLEM AND CALIBRATION OF PARAMETERS
INVERSE PROBLEM AND CALIBRATION OF PARAMETERS PART 1: An example of inverse problem: Quantification of the uncertainties of the physical models of the CATHARE code with the CIRCÉ method 1. Introduction
More informationNumerical and Experimental Results
(G) Numerical and Experimental Results To begin, we shall show some data of R. D. Moser, J. Kim & N. N. Mansour, Direct numerical simulation of turbulent flow up to Re τ = 59, Phys. Fluids 11 943-945 (1999)
More informationOn the Assessment of a Bayesian Validation Methodology for Data Reduction Models Relevant to Shock Tube Experiments
ICES REPORT -36 November 2 On the Assessment of a Bayesian Validation Methodology for Data Reduction Models Relevant to Shock Tube Experiments by M. Panesi, K. Miki, S. Prudhomme, and A. Brandis The Institute
More informationUncertainty Quantification in Performance Evaluation of Manufacturing Processes
Uncertainty Quantification in Performance Evaluation of Manufacturing Processes Manufacturing Systems October 27, 2014 Saideep Nannapaneni, Sankaran Mahadevan Vanderbilt University, Nashville, TN Acknowledgement
More informationPractical Statistics
Practical Statistics Lecture 1 (Nov. 9): - Correlation - Hypothesis Testing Lecture 2 (Nov. 16): - Error Estimation - Bayesian Analysis - Rejecting Outliers Lecture 3 (Nov. 18) - Monte Carlo Modeling -
More informationZonal hybrid RANS-LES modeling using a Low-Reynolds-Number k ω approach
Zonal hybrid RANS-LES modeling using a Low-Reynolds-Number k ω approach S. Arvidson 1,2, L. Davidson 1, S.-H. Peng 1,3 1 Chalmers University of Technology 2 SAAB AB, Aeronautics 3 FOI, Swedish Defence
More informationUncertainty Management and Quantification in Industrial Analysis and Design
Uncertainty Management and Quantification in Industrial Analysis and Design www.numeca.com Charles Hirsch Professor, em. Vrije Universiteit Brussel President, NUMECA International The Role of Uncertainties
More informationUse of Bayesian multivariate prediction models to optimize chromatographic methods
Use of Bayesian multivariate prediction models to optimize chromatographic methods UCB Pharma! Braine lʼalleud (Belgium)! May 2010 Pierre Lebrun, ULg & Arlenda Bruno Boulanger, ULg & Arlenda Philippe Lambert,
More informationFundamentals of Fluid Dynamics: Elementary Viscous Flow
Fundamentals of Fluid Dynamics: Elementary Viscous Flow Introductory Course on Multiphysics Modelling TOMASZ G. ZIELIŃSKI bluebox.ippt.pan.pl/ tzielins/ Institute of Fundamental Technological Research
More informationSpatial Statistics with Image Analysis. Outline. A Statistical Approach. Johan Lindström 1. Lund October 6, 2016
Spatial Statistics Spatial Examples More Spatial Statistics with Image Analysis Johan Lindström 1 1 Mathematical Statistics Centre for Mathematical Sciences Lund University Lund October 6, 2016 Johan Lindström
More information7.6 Example von Kármán s Laminar Boundary Layer Problem
CEE 3310 External Flows (Boundary Layers & Drag, Nov. 11, 2016 157 7.5 Review Non-Circular Pipes Laminar: f = 64/Re DH ± 40% Turbulent: f(re DH, ɛ/d H ) Moody chart for f ± 15% Bernoulli-Based Flow Metering
More informationAt A Glance. UQ16 Mobile App.
At A Glance UQ16 Mobile App Scan the QR code with any QR reader and download the TripBuilder EventMobile app to your iphone, ipad, itouch or Android mobile device. To access the app or the HTML 5 version,
More informationEstimation of Operational Risk Capital Charge under Parameter Uncertainty
Estimation of Operational Risk Capital Charge under Parameter Uncertainty Pavel V. Shevchenko Principal Research Scientist, CSIRO Mathematical and Information Sciences, Sydney, Locked Bag 17, North Ryde,
More informationStatistical Methods in Particle Physics
Statistical Methods in Particle Physics Lecture 11 January 7, 2013 Silvia Masciocchi, GSI Darmstadt s.masciocchi@gsi.de Winter Semester 2012 / 13 Outline How to communicate the statistical uncertainty
More informationScalable algorithms for optimal experimental design for infinite-dimensional nonlinear Bayesian inverse problems
Scalable algorithms for optimal experimental design for infinite-dimensional nonlinear Bayesian inverse problems Alen Alexanderian (Math/NC State), Omar Ghattas (ICES/UT-Austin), Noémi Petra (Applied Math/UC
More informationBOUNDARY LAYER ANALYSIS WITH NAVIER-STOKES EQUATION IN 2D CHANNEL FLOW
Proceedings of,, BOUNDARY LAYER ANALYSIS WITH NAVIER-STOKES EQUATION IN 2D CHANNEL FLOW Yunho Jang Department of Mechanical and Industrial Engineering University of Massachusetts Amherst, MA 01002 Email:
More informationLinear Regression Models
Linear Regression Models Model Description and Model Parameters Modelling is a central theme in these notes. The idea is to develop and continuously improve a library of predictive models for hazards,
More informationElectroweak Physics at the Tevatron
Electroweak Physics at the Tevatron Adam Lyon / Fermilab for the DØ and CDF collaborations 15 th Topical Conference on Hadron Collider Physics June 2004 Outline Importance Methodology Single Boson Measurements
More informationParametric Techniques Lecture 3
Parametric Techniques Lecture 3 Jason Corso SUNY at Buffalo 22 January 2009 J. Corso (SUNY at Buffalo) Parametric Techniques Lecture 3 22 January 2009 1 / 39 Introduction In Lecture 2, we learned how to
More informationRobust MCMC Sampling with Non-Gaussian and Hierarchical Priors
Division of Engineering & Applied Science Robust MCMC Sampling with Non-Gaussian and Hierarchical Priors IPAM, UCLA, November 14, 2017 Matt Dunlop Victor Chen (Caltech) Omiros Papaspiliopoulos (ICREA,
More informationA Probabilistic Framework for solving Inverse Problems. Lambros S. Katafygiotis, Ph.D.
A Probabilistic Framework for solving Inverse Problems Lambros S. Katafygiotis, Ph.D. OUTLINE Introduction to basic concepts of Bayesian Statistics Inverse Problems in Civil Engineering Probabilistic Model
More informationNONLINEAR FEATURES IN EXPLICIT ALGEBRAIC MODELS FOR TURBULENT FLOWS WITH ACTIVE SCALARS
June - July, 5 Melbourne, Australia 9 7B- NONLINEAR FEATURES IN EXPLICIT ALGEBRAIC MODELS FOR TURBULENT FLOWS WITH ACTIVE SCALARS Werner M.J. Lazeroms () Linné FLOW Centre, Department of Mechanics SE-44
More informationROLE OF LARGE SCALE MOTIONS IN TURBULENT POISEUILLE AND COUETTE FLOWS
10 th International Symposium on Turbulence and Shear Flow Phenomena (TSFP10), Chicago, USA, July, 2017 ROLE OF LARGE SCALE MOTIONS IN TURBULENT POISEUILLE AND COUETTE FLOWS Myoungkyu Lee Robert D. Moser
More informationAA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS
AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 1 / 29 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS Hierarchy of Mathematical Models 1 / 29 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 2 / 29
More informationAn Overview of ASME V&V 20: Standard for Verification and Validation in Computational Fluid Dynamics and Heat Transfer
An Overview of ASME V&V 20: Standard for Verification and Validation in Computational Fluid Dynamics and Heat Transfer Hugh W. Coleman Department of Mechanical and Aerospace Engineering University of Alabama
More informationStochastic Spectral Approaches to Bayesian Inference
Stochastic Spectral Approaches to Bayesian Inference Prof. Nathan L. Gibson Department of Mathematics Applied Mathematics and Computation Seminar March 4, 2011 Prof. Gibson (OSU) Spectral Approaches to
More informationA Simple Turbulence Closure Model
A Simple Turbulence Closure Model Atmospheric Sciences 6150 1 Cartesian Tensor Notation Reynolds decomposition of velocity: Mean velocity: Turbulent velocity: Gradient operator: Advection operator: V =
More informationTurbulent Boundary Layers & Turbulence Models. Lecture 09
Turbulent Boundary Layers & Turbulence Models Lecture 09 The turbulent boundary layer In turbulent flow, the boundary layer is defined as the thin region on the surface of a body in which viscous effects
More information2D Image Processing (Extended) Kalman and particle filter
2D Image Processing (Extended) Kalman and particle filter Prof. Didier Stricker Dr. Gabriele Bleser Kaiserlautern University http://ags.cs.uni-kl.de/ DFKI Deutsches Forschungszentrum für Künstliche Intelligenz
More informationTurbulence Modeling. Cuong Nguyen November 05, The incompressible Navier-Stokes equations in conservation form are u i x i
Turbulence Modeling Cuong Nguyen November 05, 2005 1 Incompressible Case 1.1 Reynolds-averaged Navier-Stokes equations The incompressible Navier-Stokes equations in conservation form are u i x i = 0 (1)
More informationBayesian model selection for computer model validation via mixture model estimation
Bayesian model selection for computer model validation via mixture model estimation Kaniav Kamary ATER, CNAM Joint work with É. Parent, P. Barbillon, M. Keller and N. Bousquet Outline Computer model validation
More informationMeasurement And Uncertainty
Measurement And Uncertainty Based on Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results, NIST Technical Note 1297, 1994 Edition PHYS 407 1 Measurement approximates or
More informationOn the validation study devoted to stratified atmospheric flow over an isolated hill
On the validation study devoted to stratified atmospheric flow over an isolated hill Sládek I. 2/, Kozel K. 1/, Jaňour Z. 2/ 1/ U1211, Faculty of Mechanical Engineering, Czech Technical University in Prague.
More informationStructural Reliability
Structural Reliability Thuong Van DANG May 28, 2018 1 / 41 2 / 41 Introduction to Structural Reliability Concept of Limit State and Reliability Review of Probability Theory First Order Second Moment Method
More informationProbing the covariance matrix
Probing the covariance matrix Kenneth M. Hanson Los Alamos National Laboratory (ret.) BIE Users Group Meeting, September 24, 2013 This presentation available at http://kmh-lanl.hansonhub.com/ LA-UR-06-5241
More informationEssentials of expressing measurement uncertainty
Essentials of expressing measurement uncertainty This is a brief summary of the method of evaluating and expressing uncertainty in measurement adopted widely by U.S. industry, companies in other countries,
More informationNumerical methods for the Navier- Stokes equations
Numerical methods for the Navier- Stokes equations Hans Petter Langtangen 1,2 1 Center for Biomedical Computing, Simula Research Laboratory 2 Department of Informatics, University of Oslo Dec 6, 2012 Note:
More informationOverview. Bayesian assimilation of experimental data into simulation (for Goland wing flutter) Why not uncertainty quantification?
Delft University of Technology Overview Bayesian assimilation of experimental data into simulation (for Goland wing flutter), Simao Marques 1. Why not uncertainty quantification? 2. Why uncertainty quantification?
More informationSTRESS TRANSPORT MODELLING 2
STRESS TRANSPORT MODELLING 2 T.J. Craft Department of Mechanical, Aerospace & Manufacturing Engineering UMIST, Manchester, UK STRESS TRANSPORT MODELLING 2 p.1 Introduction In the previous lecture we introduced
More informationStochastic Elastic-Plastic Finite Element Method for Performance Risk Simulations
Stochastic Elastic-Plastic Finite Element Method for Performance Risk Simulations Boris Jeremić 1 Kallol Sett 2 1 University of California, Davis 2 University of Akron, Ohio ICASP Zürich, Switzerland August
More informationBayesian analysis in nuclear physics
Bayesian analysis in nuclear physics Ken Hanson T-16, Nuclear Physics; Theoretical Division Los Alamos National Laboratory Tutorials presented at LANSCE Los Alamos Neutron Scattering Center July 25 August
More informationStandard Errors & Confidence Intervals. N(0, I( β) 1 ), I( β) = [ 2 l(β, φ; y) β i β β= β j
Standard Errors & Confidence Intervals β β asy N(0, I( β) 1 ), where I( β) = [ 2 l(β, φ; y) ] β i β β= β j We can obtain asymptotic 100(1 α)% confidence intervals for β j using: β j ± Z 1 α/2 se( β j )
More informationPreliminary Statistics Lecture 2: Probability Theory (Outline) prelimsoas.webs.com
1 School of Oriental and African Studies September 2015 Department of Economics Preliminary Statistics Lecture 2: Probability Theory (Outline) prelimsoas.webs.com Gujarati D. Basic Econometrics, Appendix
More informationAASS, Örebro University
Mobile Robotics Achim J. and Lilienthal Olfaction Lab, AASS, Örebro University 1 Contents 1. Gas-Sensitive Mobile Robots in the Real World 2. Gas Dispersal in Natural Environments 3. Gas Quantification
More informationRiemann Manifold Methods in Bayesian Statistics
Ricardo Ehlers ehlers@icmc.usp.br Applied Maths and Stats University of São Paulo, Brazil Working Group in Statistical Learning University College Dublin September 2015 Bayesian inference is based on Bayes
More informationStatistics. Lecture 4 August 9, 2000 Frank Porter Caltech. 1. The Fundamentals; Point Estimation. 2. Maximum Likelihood, Least Squares and All That
Statistics Lecture 4 August 9, 2000 Frank Porter Caltech The plan for these lectures: 1. The Fundamentals; Point Estimation 2. Maximum Likelihood, Least Squares and All That 3. What is a Confidence Interval?
More informationModel comparison. Patrick Breheny. March 28. Introduction Measures of predictive power Model selection
Model comparison Patrick Breheny March 28 Patrick Breheny BST 760: Advanced Regression 1/25 Wells in Bangladesh In this lecture and the next, we will consider a data set involving modeling the decisions
More informationDimension-Independent likelihood-informed (DILI) MCMC
Dimension-Independent likelihood-informed (DILI) MCMC Tiangang Cui, Kody Law 2, Youssef Marzouk Massachusetts Institute of Technology 2 Oak Ridge National Laboratory 2 August 25 TC, KL, YM DILI MCMC USC
More informationThe Role of Uncertainty Quantification in Model Verification and Validation. Part 2 Model Verification and Validation Plan and Process
The Role of Uncertainty Quantification in Model Verification and Validation Part 2 Model Verification and Validation Plan and Process 1 Case Study: Ni Superalloy Yield Strength Model 2 3 What is to be
More informationData assimilation as an optimal control problem and applications to UQ
Data assimilation as an optimal control problem and applications to UQ Walter Acevedo, Angwenyi David, Jana de Wiljes & Sebastian Reich Universität Potsdam/ University of Reading IPAM, November 13th 2017
More informationSession: Probabilistic methods in avalanche hazard assessment. Session moderator: Marco Uzielli Norwegian Geotechnical Institute
Session: Probabilistic methods in avalanche hazard assessment Session moderator: Marco Uzielli Norwegian Geotechnical Institute Session plan Title Presenter Affiliation A probabilistic framework for avalanche
More informationNumerical Methods in Aerodynamics. Turbulence Modeling. Lecture 5: Turbulence modeling
Turbulence Modeling Niels N. Sørensen Professor MSO, Ph.D. Department of Civil Engineering, Alborg University & Wind Energy Department, Risø National Laboratory Technical University of Denmark 1 Outline
More informationMelbourne, Victoria, 3010, Australia. To link to this article:
This article was downloaded by: [The University Of Melbourne Libraries] On: 18 October 2012, At: 21:54 Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: 1072954
More informationStep-Stress Models and Associated Inference
Department of Mathematics & Statistics Indian Institute of Technology Kanpur August 19, 2014 Outline Accelerated Life Test 1 Accelerated Life Test 2 3 4 5 6 7 Outline Accelerated Life Test 1 Accelerated
More informationElliptic relaxation for near wall turbulence models
Elliptic relaxation for near wall turbulence models J.C. Uribe University of Manchester School of Mechanical, Aerospace & Civil Engineering Elliptic relaxation for near wall turbulence models p. 1/22 Outline
More informationAN UNCERTAINTY ESTIMATION EXAMPLE FOR BACKWARD FACING STEP CFD SIMULATION. Abstract
nd Workshop on CFD Uncertainty Analysis - Lisbon, 19th and 0th October 006 AN UNCERTAINTY ESTIMATION EXAMPLE FOR BACKWARD FACING STEP CFD SIMULATION Alfredo Iranzo 1, Jesús Valle, Ignacio Trejo 3, Jerónimo
More informationHybrid LES RANS Method Based on an Explicit Algebraic Reynolds Stress Model
Hybrid RANS Method Based on an Explicit Algebraic Reynolds Stress Model Benoit Jaffrézic, Michael Breuer and Antonio Delgado Institute of Fluid Mechanics, LSTM University of Nürnberg bjaffrez/breuer@lstm.uni-erlangen.de
More informationPredictive Engineering and Computational Sciences. Local Sensitivity Derivative Enhanced Monte Carlo Methods. Roy H. Stogner, Vikram Garg
PECOS Predictive Engineering and Computational Sciences Local Sensitivity Derivative Enhanced Monte Carlo Methods Roy H. Stogner, Vikram Garg Institute for Computational Engineering and Sciences The University
More informationBayesian Estimation of Input Output Tables for Russia
Bayesian Estimation of Input Output Tables for Russia Oleg Lugovoy (EDF, RANE) Andrey Polbin (RANE) Vladimir Potashnikov (RANE) WIOD Conference April 24, 2012 Groningen Outline Motivation Objectives Bayesian
More informationSoil Uncertainty and Seismic Ground Motion
Boris Jeremić and Kallol Sett Department of Civil and Environmental Engineering University of California, Davis GEESD IV May 2008 Support by NSF CMMI 0600766 Outline Motivation Soils and Uncertainty Stochastic
More informationAssessment of an isolation condenser performance in case of a LOHS using the RMPS+ Methodology
Assessment of an isolation condenser performance in case of a LOHS using the RMPS+ Methodology M. Giménez, F. Mezio, P. Zanocco, G. Lorenzo Nuclear Safety Group - Centro Atómico Bariloche, CNEA - Argentina
More informationPhysics of turbulent flow
ECL-MOD 3A & MSc. Physics of turbulent flow Christophe Bailly Université de Lyon, Ecole Centrale de Lyon & LMFA - UMR CNRS 5509 http://acoustique.ec-lyon.fr Outline of the course A short introduction to
More informationOn modeling pressure diusion. in non-homogeneous shear ows. By A. O. Demuren, 1 M. M. Rogers, 2 P. Durbin 3 AND S. K. Lele 3
Center for Turbulence Research Proceedings of the Summer Program 1996 63 On modeling pressure diusion in non-homogeneous shear ows By A. O. Demuren, 1 M. M. Rogers, 2 P. Durbin 3 AND S. K. Lele 3 New models
More informationStatistical Measures of Uncertainty in Inverse Problems
Statistical Measures of Uncertainty in Inverse Problems Workshop on Uncertainty in Inverse Problems Institute for Mathematics and Its Applications Minneapolis, MN 19-26 April 2002 P.B. Stark Department
More informationTurbulent Flows. g u
.4 g u.3.2.1 t. 6 4 2 2 4 6 Figure 12.1: Effect of diffusion on PDF shape: solution to Eq. (12.29) for Dt =,.2,.2, 1. The dashed line is the Gaussian with the same mean () and variance (3) as the PDF at
More informationThe Bayesian approach to inverse problems
The Bayesian approach to inverse problems Youssef Marzouk Department of Aeronautics and Astronautics Center for Computational Engineering Massachusetts Institute of Technology ymarz@mit.edu, http://uqgroup.mit.edu
More informationProbabilistic Graphical Models Homework 2: Due February 24, 2014 at 4 pm
Probabilistic Graphical Models 10-708 Homework 2: Due February 24, 2014 at 4 pm Directions. This homework assignment covers the material presented in Lectures 4-8. You must complete all four problems to
More informationRELIABILITY, UNCERTAINTY, ESTIMATES VALIDATION AND VERIFICATION
RELIABILITY, UNCERTAINTY, ESTIMATES VALIDATION AND VERIFICATION Transcription of I. Babuska s presentation at the workshop on the Elements of Predictability, J. Hopkins Univ, Baltimore, MD, Nov. 13-14,
More information